Vinyl floor tiles curl up at the corners as their adhesive starts to fail, but there's usually still enough stick to them that you can't just pull them up without damage. You can use a simple clothes iron to loosen that remaining adhesive.
As DIY Network shows us, all you have to do is place a sheet of aluminium foil down to protect the surface of the tile, then warm it up with an iron on medium heat. The heat will soften the adhesive and make the tile more flexible. Pull the tile up enough to spread new adhesive, then press it back down. Hit up the link for the full instructions.